We derive inequalities for the probabilities of projective measurements in mutually unbiased bases of a qudit system. These inequalities lead to wider ranges of validity and tighter bounds on entropic uncertainty inequalities previously derived in the literature.
We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In particular, we show that a bound derived from the result by Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] for two such measurements can in fact be tight for up to d measurements in mutually unbiased bases. We then show that using more mutually unbiased bases does not always lead to a better locking effect. We prove that the optimal bound for the accessible information using up to d specific mutually unbiased bases is logd∕2, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect and we need to look for additional properties.
In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives no information about the outcomes of measurements in another basis. Here we relax the no-information condition by allowing the d outcomes to be predicted according to a predefined probability distribution $q=({q}_{0},...,{q}_{d\ensuremath{-}1})$. The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of mutually equibiased bases (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For d=2,3, we derive a set of d+1 MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution q, leading, for d=3, to the restriction $1/3\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1/2$, where $\ensuremath{\mu}={\ensuremath{\sum}}_{k=0}^{2}{q}_{k}^{2}$. To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index $\ensuremath{\mu}$, employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a $3\ifmmode\times\else\texttimes\fi{}3$ system. Our results reveal that not all bases in a set of L MEBs can contribute to the entanglement detection. A constraint, dependent on the probability distribution q, limits the number of MEBs that can be employed.
The current study aims to examine uncertainty relations for quantum measurements assigned to a projective two-design. Complete sets of mutually unbiased bases and symmetric informationally complete measurements are important cases of such measurements. To characterize the amount of uncertainty, we use the Tsallis and Rényi entropies as well as the probabilities of separate outcomes. The obtained results are based on an estimation of the index of coincidence. They improve some uncertainty relations given in the literature.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
We investigate the fine-grained uncertainty relations for qubit system by measurements corresponding to two and three spin operators. Then we derive the general bound for a combination of two probabilities of projective measurements in mutually unbiased bases in d-dimensional Hilbert space (d is prime). Those uncertainty inequalities can be applied to construct different thermodynamic cycles which all imply the second law of thermodynamics.